I searched it on the internets but it seems like it's so common knowledge that there are no articles or explanations about it.
From a logical gut-feeling kind of guess, I would say no.
Any confirmation would be quite appreciated.
There are a few dozen publicly traded brokerage firms in the US.
Some of the better known names include E*TRADE, Charles Schwab, TD Ameritrade, Interactive Brokers, Goldman Sachs, Raymond James...
they are publicly traded companies.
TD ameritrade (a large stock brokerage house)
Not only are many brokerages publicly traded companies, so are some of the exchanges.
The NYSE is owned by Intercontinental Exchange Inc with the symbol ICE which is traded on the NYSE and a member of the S&P500.
NASDAQ is owned by NASDAQ Inc with the stock symbol NDAQ it is listed on the NASDAQ and also part of the S&P500.